The size function of quadratic extensions of complex quadratic fields

TL;DR

This paper proves a conjecture regarding the maximality of the size function $h^0$ at the trivial Arakelov divisor specifically for quadratic extensions of complex quadratic fields, advancing understanding in number theory.

Contribution

It confirms van der Geer and Schoof's conjecture, establishing the maximality of $h^0$ at the trivial divisor for these specific quadratic extensions.

Findings

$h^0$ attains its maximum at the trivial Arakelov divisor for quadratic extensions of complex quadratic fields

The result supports the analogy between $h^0$ and Riemann-Roch dimensions in algebraic geometry

Advances the understanding of Arakelov invariants in number field extensions.

Abstract

The function $h^0$ for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. In this paper, we prove the conjecture of van der Geer and Schoof about the maximality of $h^0$ at the trivial Arakelov divisor for quadratic extensions of complex quadratic fields.